Properties

Label 1.27.ak
Base Field $\F_{3^{3}}$
Dimension $1$
Ordinary No
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{3^{3}}$
Dimension:  $1$
Weil polynomial:  $1 - 10 x + 27 x^{2}$
Frobenius angles:  $\pm0.0877398280459$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-2}) \)
Galois group:  $C_2$
Jacobians:  1

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $1$
Slopes:  $[0, 1]$

Point counts

This isogeny class contains the Jacobians of 1 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 18 684 19494 530784 14347458 387423756 10460425014 282430166400 7625601845298 205891158689964

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 18 684 19494 530784 14347458 387423756 10460425014 282430166400 7625601845298 205891158689964

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{3}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-2}) \).
All geometric endomorphisms are defined over $\F_{3^{3}}$.

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{3^{3}}$.

SubfieldPrimitive Model
$\F_{3}$1.3.c

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
1.27.k$2$1.729.abu